Question: All of the 3rd grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$3.00$ each for students, and the group paid $$42.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$22.50$ each for teachers and $$7.50$ each for students, and the group paid $$112.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3y = 42}$ ${22.5x+7.5y = 112.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-22.5x-9y = -126}$ ${22.5x+7.5y = 112.5}$ Add the top and bottom equations together. $ -1.5y = -13.5 $ $ y = \dfrac{-13.5}{-1.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {7.5x+3y = 42}$ to find $x$ ${7.5x + 3}{(9)}{= 42}$ $7.5x+27 = 42$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {22.5x+7.5y = 112.5}$ and get the same answer for $x$ ${22.5x + 7.5}{(9)}{= 112.5}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.